A156016 Expansion of (1-x-sqrt(1-6x-3x^2))/(2x).

1, 3, 9, 36, 162, 783, 3969, 20817, 112023, 615033, 3431403, 19398690, 110880900, 639730305, 3720657807, 21790419444, 128398625658, 760668489729, 4528069760691, 27070491820644, 162464919528222, 978463778897637

Offset: 0

Author

Paul Barry, Feb 01 2009

Keywords

• nonn
• easy

Comments

Transform of Catalan numbers by Riordan array ((1+x)/(1-x), x(1+x)/(1-x)^2).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
• M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.

Crossrefs

Cf. A000108, A100239, A107264.

Programs

Mathematica

CoefficientList[Series[(1-x-Sqrt[1-6x-3x^2])/(2x),{x,0,30}],x] (* Harvey P. Dale, Jul 27 2014 *)

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..k+1} C(k+1,j)*C(n+k-j,n-k-j)*A000108(k).

a(n+1) = 3*A107264(n-1). - Philippe Deléham, Feb 04 2009

D-finite with recurrence: (n+1)*a(n) + 3*(-2*n+1)*a(n-1) + 3*(-n+2)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014

G.f. A(x) satisfies: A(x) = 1 + x * (1 + A(x) + A(x)^2). - Ilya Gutkovskiy, Jul 01 2020